- Research
- Open access
- Published:
Unitarily invariant norm inequalities involving Heron and Heinz means
Journal of Inequalities and Applications volumeĀ 2014, ArticleĀ number:Ā 288 (2014)
Abstract
In this paper, we present some new inequalities for unitarily invariant norms involving Heron and Heinz means for matrices, which generalize the result of TheoremĀ 2.1 (Fu and He in J. Math. Inequal. 7(4):727-737, 2013) and refine the inequality of TheoremĀ 6 (Zhan in SIAM J. Matrix Anal. Appl. 20: 466-470, 1998). Our results are a refinement and a generalization of some existing inequalities.
MSC:47A30, 15A60.
1 Introduction
Throughout, let be the space of complex matrices and .
A norm is called unitarily invariant norm if for all and for all unitary matrices . Two classes of unitarily invariant norms are especially important. The first is the class of the Ky Fan k-norm , defined as
where () are the singular values of A with , which are the eigenvalues of the positive semidefinite matrix , arranged in decreasing order and repeated according to multiplicity. The second is the class of the Schatten p-norm , defined as
For two nonnegative real numbers a and b, the Heinz mean and Heron mean in the parameter v, , are defined, respectively, as
Note that (the arithmetic mean of a and b) and (the geometric mean of a and b). It is easy to see that as a function of v, is convex, attains its minimum at , and attains its maximum at and .
The operator version of the Heinz mean [1] asserts that if A, B and X are operators on a complex separable Hilbert space such that A and B are positive, then for every unitarily invariant norm , the function is convex on , attains its minimum at , and attains its maximum at and . Moreover, the operator version of the Heron mean [2] asserts that .
Let , A, B are positive definite, Kaur and Singh [2] have proved the following inequalities for any unitarily invariant norm :
and
where , and .
Replacing A, B by , in (1.1) and (1.2), then putting , the following inequalities hold:
and
where , and .
Zhan proved in [3] that if , such that A, B are positive semidefinite, then
for and .
Let , such that A, B are positive semidefinite, for and ; Fu et al. in [4] proved that
Recently, Kaur et al. [5], He et al. [6] and Bakherad et al. [7] have studied similar topics.
For the sake of convenience, we set
In Section 2, we will generalize and refine some existing inequalities for unitarily invariant norms involving Heron and Heinz means for matrices and present some new refinements of the inequalities above.
2 Main results
In this section, we firstly utilize the convexity of the function to obtain a unitarily invariant norms inequality that leads to another version of the inequality (1.6), which is also the refinement of the inequality (1.5).
To obtain the results, we need the following lemma on convex functions [8, 9].
Lemma 2.1 Let f be a real valued continuous convex function on an interval which contains . Then for , we have
Theorem 2.2 Let , such that A, B are positive semidefinite. Then for any unitarily invariant norm , and ,
where .
Proof For , by the convexity of the function and Lemma 2.1, presented above, we have
which implies
By (1.3) and (2.2), we have
So,
For , by the convexity of the function and Lemma 2.1, presented above, we have
which implies
By (1.3) and (2.4), we have
So,
By (2.3) and (2.5), for , and , we have the following equivalent inequality:
The proof is completed.āā”
Remark 2.3 With a simple computation between the upper bounds in (1.3) and (2.1), obviously we have
Thus the inequality (2.1) is a refinement of the inequality (1.3).
Now, we present a refinement of the inequality .
Theorem 2.4 Let , such that A, B are positive semidefinite. Then for any unitarily invariant norm , , and , we have
where .
Proof For , from Theorem 2.2, we have
By integrating both sides of the inequality above, we have
For , from Theorem 2.2, we have
Similarly, by integrating both sides of the inequality above, we have
It follows from (2.7) and (2.8) that
which is equivalent to
The proof is completed.āā”
Remark 2.5 Obviously,
Thus, the inequality (2.6) is a refinement of the inequality .
Taking (), the following corollaries are obtained.
Corollary 2.6 Let , such that A, B are positive semidefinite. Then for any unitarily invariant norm and ,
where and .
Corollary 2.7 Let , such that A, B are positive semidefinite. Then for any unitarily invariant norm , ,
where and .
Thus, on the one hand, the inequality (2.9) is a refinement of the inequality , and also another version of the inequality (1.6); on the other hand, the inequality (2.10) is just the inequality proved in [4], so the inequality (2.6) presented in Theorem 2.4 is also the generalization of the inequality proved in [4].
References
Kittaneh F: On the convexity of the Heinz means. Integral Equ. Oper. Theory 2010, 68: 519ā527. 10.1007/s00020-010-1807-6
Kaur R, Singh M: Complete interpolation of matrix versions of Heron and Heinz means. Math. Inequal. Appl. 2013, 16: 93ā99.
Zhan X: Inequalities for unitarily invariant norms. SIAM J. Matrix Anal. Appl. 1998, 20: 466ā470. 10.1137/S0895479898323823
Fu X, He C: On some inequalities for unitarily invariant norms. J. Math. Inequal. 2013,7(4):727ā737.
Kaur R, Moslehian MS, Singh M, Conde C: Further refinements of the Heinz inequality. Linear Algebra Appl. 2014, 447: 26ā37.
He CJ, Zou LM, Qaisar S: On improved arithmetric-geometric mean and Heinz inequalities for matrices. J. Math. Inequal. 2012,6(3):453ā459.
Bakherad M, Moslehian MS: Reverses and variations of Heinz inequality. Linear Multilinear Algebra 2014. 10.1080/03081087.2014.880433
Bhatia R, Sharma R: Some inequalities for positive linear maps. Linear Algebra Appl. 2012, 436: 1562ā1571. 10.1016/j.laa.2010.09.038
Wang S, Zou L, Jiang Y: Some inequalities for unitarily invariant norms of matrices. J. Inequal. Appl. 2011., 2011: Article ID 10
Acknowledgements
The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests. HC is responsible for all the whole of the article appearing.
Authorsā contributions
JW carried out the matrix operator theory studies and participated in the conception and design. HC conceived of the study, participated in its design and drafting the manuscript. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Cao, H., Wu, J. Unitarily invariant norm inequalities involving Heron and Heinz means. J Inequal Appl 2014, 288 (2014). https://doi.org/10.1186/1029-242X-2014-288
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-288